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Alternating sum of inverse of increasing integers with a difference of 0, 1, 2, 3, ...: 1 - 1/2 + 1/4 - 1/7 + 1/11 - 1/16 + 1/22 - 1/29 + 1/37 + ... i.e., alternating series based on A226985.
+20
2
6, 6, 1, 5, 7, 0, 1, 9, 2, 0, 7, 3, 5, 8, 5, 1, 1, 2, 0, 4, 4, 5, 7, 3, 8, 9, 2, 8, 4, 6, 0, 7, 9, 3, 9, 5, 2, 1, 7, 6, 4, 2, 4, 6, 6, 5, 8, 9, 5, 5, 6, 9, 7, 9, 8, 6, 9, 1, 9, 8, 4, 8, 5, 4, 5, 0, 1, 8, 9, 5, 0, 9, 7, 9, 4, 2, 6, 0, 1, 7, 2, 0, 7, 5, 9, 5, 8, 8, 8, 7, 7, 9, 1, 1, 8, 6, 9, 3, 7, 2, 4, 4, 9, 2, 7, 9, 4, 8
EXAMPLE
0.66157019207358511204457389...
MAPLE
c:= Sum( (-1)^k/(1+binomial(k+1, 2)), k=0..infinity):
MATHEMATICA
N[((-2 I) (LerchPhi[-1, 1, 1/2 - (I/2) Sqrt[7]] - LerchPhi[-1, 1, 1/2 + (I/2) Sqrt[7]]))/Sqrt[7], 99] (* Joerg Arndt, Sep 09 2013 *) -(2*Im[PolyGamma[(1-I*Sqrt[7])/4] - PolyGamma[(3-I*Sqrt[7])/4]])/Sqrt[7] // RealDigits[#, 10, 100]& // First (* Jean-François Alcover, Sep 10 2013 *)
PROG
(PARI) default(realprecision, 133); sumalt(k=1, 1/(1+k*(k-1)/2)*(-1)^(k+1))
(PARI) -(2*imag(psi((1-I*sqrt(7))/4)-psi((3-I*sqrt(7))/4)))/sqrt(7) \\ sumalt is faster; Charles R Greathouse IV, Sep 10 2013
Central polygonal numbers (the Lazy Caterer's sequence): n(n+1)/2 + 1; or, maximal number of pieces formed when slicing a pancake with n cuts.
(Formerly M1041 N0391)
+10
430
1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, 172, 191, 211, 232, 254, 277, 301, 326, 352, 379, 407, 436, 466, 497, 529, 562, 596, 631, 667, 704, 742, 781, 821, 862, 904, 947, 991, 1036, 1082, 1129, 1177, 1226, 1276, 1327, 1379
COMMENTS
These are Hogben's central polygonal numbers with the (two-dimensional) symbol
2
.P
1 n
The first line cuts the pancake into 2 pieces. For n > 1, the n-th line crosses every earlier line (avoids parallelism) and also avoids every previous line intersection, thus increasing the number of pieces by n. For 16 lines, for example, the number of pieces is 2 + 2 + 3 + 4 + 5 + ... + 16 = 137. These are the triangular numbers plus 1 (cf. A000217). m = (n-1)(n-2)/2 + 1 is also the smallest number of edges such that all graphs with n nodes and m edges are connected. - Keith Briggs, May 14 2004 Also maximal number of grandchildren of a binary vector of length n+2. E.g., a binary vector of length 6 can produce at most 11 different vectors when 2 bits are deleted.
This is also the order dimension of the (strong) Bruhat order on the finite Coxeter group B_{n+1}. - Nathan Reading (reading(AT)math.umn.edu), Mar 07 2002
Number of 132- and 321-avoiding permutations of {1,2,...,n+1}. - Emeric Deutsch, Mar 14 2002 For n >= 1 a(n) is the number of terms in the expansion of (x+y)*(x^2+y^2)*(x^3+y^3)*...*(x^n+y^n). - Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 28 2003
Also the number of terms in (1)(x+1)(x^2+x+1)...(x^n+...+x+1); see A000140. Narayana transform (analog of the binomial transform) of vector [1, 1, 0, 0, 0, ...] = A000124; using the infinite lower Narayana triangle of A001263 (as a matrix), N; then N * [1, 1, 0, 0, 0, ...] = A000124. - Gary W. Adamson, Apr 28 2005 Number of interval subsets of {1, 2, 3, ..., n} (cf. A002662). - Jose Luis Arregui (arregui(AT)unizar.es), Jun 27 2006 Define a number of straight lines in the plane to be in general arrangement when (1) no two lines are parallel, (2) there is no point common to three lines. Then these are the maximal numbers of regions defined by n straight lines in general arrangement in the plane. - Peter C. Heinig (algorithms(AT)gmx.de), Oct 19 2006
Note that a(n) = a(n-1) + A000027(n-1). This has the following geometrical interpretation: Suppose there are already n-1 lines in general arrangement, thus defining the maximal number of regions in the plane obtainable by n-1 lines and now one more line is added in general arrangement. Then it will cut each of the n-1 lines and acquire intersection points which are in general arrangement. (See the comments on A000027 for general arrangement with points.) These points on the new line define the maximal number of regions in 1-space definable by n-1 points, hence this is A000027(n-1), where for A000027 an offset of 0 is assumed, that is, A000027(n-1) = (n+1)-1 = n. Each of these regions acts as a dividing wall, thereby creating as many new regions in addition to the a(n-1) regions already there, hence a(n) = a(n-1) + A000027(n-1). Cf. the comments on A000125 for an analogous interpretation. - Peter C. Heinig (algorithms(AT)gmx.de), Oct 19 2006 When constructing a zonohedron, one zone at a time, out of (up to) 3-d non-intersecting parallelepipeds, the n-th element of this sequence is the number of edges in the n-th zone added with the n-th "layer" of parallelepipeds. (Verified up to 10-zone zonohedron, the enneacontahedron.) E.g., adding the 10th zone to the enneacontahedron requires 46 parallel edges (edges in the 10th zone) by looking directly at a 5-valence vertex and counting visible vertices. - Shel Kaphan, Feb 16 2006 Binomial transform of (1, 1, 1, 0, 0, 0, ...) and inverse binomial transform of A072863: (1, 3, 9, 26, 72, 192, ...). - Gary W. Adamson, Oct 15 2007 If Y is a 2-subset of an n-set X then, for n >= 3, a(n-3) is the number of (n-2)-subsets of X which do not have exactly one element in common with Y. - Milan Janjic, Dec 28 2007 It appears that a(n) is the number of distinct values among the fractions F(i+1)/F(j+1) as j ranges from 1 to n and, for each fixed j, i ranges from 1 to j, where F(i) denotes the i-th Fibonacci number. - John W. Layman, Dec 02 2008 a(n) is the number of subsets of {1,2,...,n} that contain at most two elements. - Geoffrey Critzer, Mar 10 2009 For n >= 2, a(n) gives the number of sets of subsets A_1, A_2, ..., A_n of n = {1, 2, ..., n} such that Meet_{i = 1..n} A_i is empty and Sum_{j in [n]} (|Meet{i = 1..n, i != j} A_i|) is a maximum. - Srikanth K S, Oct 22 2009 The numbers along the left edge of Floyd's triangle. - Paul Muljadi, Jan 25 2010 Let A be the Hessenberg matrix of order n, defined by: A[1,j] = A[i,i]:=1, A[i,i-1] = -1, and A[i,j] = 0 otherwise. Then, for n >= 1, a(n-1) = (-1)^(n-1)*coeff(charpoly(A,x),x). - Milan Janjic, Jan 24 2010 Also the number of deck entries of Euler's ship. See the Meijer-Nepveu link. - Johannes W. Meijer, Jun 21 2010 (1 + x^2 + x^3 + x^4 + x^5 + ...)*(1 + 2x + 3x^2 + 4x^3 + 5x^4 + ...) = (1 + 2x + 4x^2 + 7x^3 + 11x^4 + ...). - Gary W. Adamson, Jul 27 2010 The number of length n binary words that have no 0-digits between any pair of consecutive 1-digits. - Jeffrey Liese, Dec 23 2010 Let b(0) = b(1) = 1; b(n) = max(b(n-1)+n-1, b(n-2)+n-2) then a(n) = b(n+1). - Yalcin Aktar, Jul 28 2011 Also number of distinct sums of 1 through n where each of those can be + or -. E.g., {1+2,1-2,-1+2,-1-2} = {3,-1,1,-3} and a(2) = 4. - Toby Gottfried, Nov 17 2011 This sequence is complete because the sum of the first n terms is always greater than or equal to a(n+1)-1. Consequently, any nonnegative number can be written as a sum of distinct terms of this sequence. See A204009, A072638. - Frank M Jackson, Jan 09 2012 The sequence is the number of distinct sums of subsets of the nonnegative integers, and its first differences are the positive integers. See A208531 for similar results for the squares. - John W. Layman, Feb 28 2012 Apparently the number of Dyck paths of semilength n+1 in which the sum of the first and second ascents add to n+1. - David Scambler, Apr 22 2013 Without 1 and 2, a(n) equals the terminus of the n-th partial sum of sequence 1, 1, 2. Explanation: 1st partial sums of 1, 1, 2 are 1, 2, 4; 2nd partial sums are 1, 3, 7; 3rd partial sums are 1, 4, 11; 4th partial sums are 1, 5, 16, etc. - Bob Selcoe, Jul 04 2013 Equivalently, numbers of the form 2*m^2+m+1, where m = 0, -1, 1, -2, 2, -3, 3, ... . - Bruno Berselli, Apr 08 2014 For n >= 2: quasi-triangular numbers; the almost-triangular numbers being A000096(n), n >= 2. Note that 2 is simultaneously almost-triangular and quasi-triangular. - Daniel Forgues, Apr 21 2015 n points in general position determine "n choose 2" lines, so A055503(n) <= a(n(n-1)/2). If n > 3, the lines are not in general position and so A055503(n) < a(n(n-1)/2). - Jonathan Sondow, Dec 01 2015 The digital root is period 9 (1, 2, 4, 7, 2, 7, 4, 2, 1), also the digital roots of centered 10-gonal numbers ( A062786), for n > 0, A133292. - Peter M. Chema, Sep 15 2016 For n >= 0, a(n) is the number of weakly unimodal sequences of length n over the alphabet {1, 2}. - Armend Shabani, Mar 10 2017 Number of sequences (e(1), ..., e(n+1)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) < e(j) != e(k). [Martinez and Savage, 2.4]
Number of sequences (e(1), ..., e(n+1)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) < e(j) and e(i) < e(k). [Martinez and Savage, 2.4]
Number of sequences (e(1), ..., e(n+1)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) >= e(j) != e(k). [Martinez and Savage, 2.4]
(End)
The odd prime factors != 7 occur in an interval of p successive terms either never or exactly twice, while 7 always occurs only once. If a prime factor p appears in a(n) and a(m) within such an interval, then n + m == -1 (mod p). When 7 divides a(n), then 2*n == -1 (mod 7). a(n) is never divisible by the prime numbers given in A003625. While all prime factors p != 7 can occur to any power, a(n) is never divisible by 7^2. The prime factors are given in A045373. The prime terms of this sequence are given in A055469. (End)
a(n-1) is the greatest sum of arch lengths for the top arches of a semi-meander with n arches. An arch length is the number of arches covered + 1.
/\ The top arch has a length of 3. /\ The top arch has a length of 3.
/ \ Both bottom arches have a //\\ The middle arch has a length of 2.
//\/\\ length of 1. ///\\\ The bottom arch has a length of 1.
Example: for n = 4, a(4-1) = a(3) = 7 /\
//\\
/\ ///\\\ 1 + 3 + 2 + 1 = 7. (End)
a(n+1) is the a(n)-th smallest positive integer that has not yet appeared in the sequence. - Matthew Malone, Aug 26 2021 For n> 0, let the n-dimensional cube {0,1}^n be, provided with the Hamming distance, d. Given an element x in {0,1}^n, a(n) is the number of elements y in {0,1}^n such that d(x, y) <= 2. Example: n = 4. (0,0,0,0), (1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1), (0,0,1,1), (0,1,0,1), (0,1,1,0), (1,0,0,1), (1,0,1,0), (1,1,0,0) are at distance <= 2 from (0,0,0,0), so a(4) = 11. - Yosu Yurramendi, Dec 10 2021 a(n) is the sum of the first three entries of row n of Pascal's triangle. - Daniel T. Martin, Apr 13 2022 a(n-1) is the number of Grassmannian permutations that avoid a pattern, sigma, where sigma is a pattern of size 3 with exactly one descent. For example, sigma is one of the patterns, {132, 213, 231, 312}. - Jessica A. Tomasko, Sep 14 2022 a(n+4) is the number of ways to tile an equilateral triangle of side length 2*n with smaller equilateral triangles of side length n and side length 1. For example, with n=2, there are 22 ways to tile an equilateral triangle of side length 4 with smaller ones of sides 2 and 1, including the one tiling with sixteen triangles of sides 1 and the one tiling with four triangles of sides 2. - Ahmed ElKhatib and Greg Dresden, Aug 19 2024
REFERENCES
Robert B. Banks, Slicing Pizzas, Racing Turtles and Further Adventures in Applied Mathematics, Princeton Univ. Press, 1999. See p. 24.
Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 72, Problem 2.
Henry Ernest Dudeney, Amusements in Mathematics, Nelson, London, 1917, page 177.
Derrick Niederman, Number Freak, From 1 to 200 The Hidden Language of Numbers Revealed, A Perigee Book, NY, 2009, p. 83.
Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.
Alain M. Robert, A Course in p-adic Analysis, Springer-Verlag, 2000; p. 213.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane, On single-deletion-correcting codes, in Codes and Designs (Columbus, OH, 2000), 273-291, Ohio State Univ. Math. Res. Inst. Publ., 10, de Gruyter, Berlin, 2002.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 98.
William Allen Whitworth, DCC Exercises in Choice and Chance, Stechert, NY, 1945, p. 30.
Akiva M. Yaglom and Isaak M. Yaglom, Challenging Mathematical Problems with Elementary Solutions. Vol. I. Combinatorial Analysis and Probability Theory. New York: Dover Publications, Inc., 1987, p. 13, #44 (First published: San Francisco: Holden-Day, Inc., 1964).
LINKS
M. F. Hasler, Interactive illustration of A000124. [Sep 06 2017: The user can choose the slices to make, but the program can suggest a set of n slices which should yield the maximum number of pieces. For n slices this obviously requires 2n endpoints, or 2n+1 if they are equally spaced, so if there are not enough "blobs", their number is accordingly increased. This is the distinction between "draw" (done when you change the slices or number of blobs by hand) and "suggest" (propose a new set of slices).] Rodica Simion and Frank W. Schmidt, Restricted permutations, European J. Combin., 6, 383-406, 1985; see Example 3.5.
FORMULA
G.f.: (1 - x + x^2)/(1 - x)^3. - Simon Plouffe in his 1992 dissertation G.f.: (1 - x^6)/((1 - x)^2*(1 - x^2)*(1 - x^3)). a(n) = a(-1 - n) for all n in Z. - Michael Somos, Sep 04 2006 Euler transform of length 6 sequence [ 2, 1, 1, 0, 0, -1]. - Michael Somos, Sep 04 2006 a(n+3) = 3*a(n+2) - 3*a(n+1) + a(n) and a(1) = 1, a(2) = 2, a(3) = 4. - Artur Jasinski, Oct 21 2008 a(n) = a(n-1) + n. E.g.f.:(1 + x + x^2/2)*exp(x). - Geoffrey Critzer, Mar 10 2009 a(n) = Sum_{k = 0..n + 1} binomial(n+1, 2(k - n)). - Paul Barry, Aug 29 2004 a(n) = binomial(n+2, 1) - 2*binomial(n+1, 1) + binomial(n+2, 2). - Zerinvary Lajos, May 12 2006 a(n) = Sum_{l_1 = 0..n + 1} Sum_{l_2 = 0..n}...Sum_{l_i = 0..n - i}...Sum_{l_n = 0..1} delta(l_1, l_2, ..., l_i, ..., l_n) where delta(l_1, l_2, ..., l_i, ..., l_n) = 0 if any l_i != l_(i+1) and l_(i+1) != 0 and delta(l_1, l_2, ..., l_i, ..., l_n) = 1 otherwise. (End)
a(n) = 2*a(n-1) - a(n-2) + 1. - Eric Werley, Jun 27 2011 E.g.f.: exp(x)*(1+x+(x^2)/2) = Q(0); Q(k) = 1+x/(1-x/(2+x-4/(2+x*(k+1)/Q(k+1)))); (continued fraction). - Sergei N. Gladkovskii, Nov 21 2011 Dirichlet g.f.: (zeta(s-2) + zeta(s-1) + 2*zeta(s))/2.
Sum_{n >= 0} 1/a(n) = 2*Pi*tanh(sqrt(7)*Pi/2)/sqrt(7) = A226985. (End) a(n) = 2*a(n-1) - binomial(n-1, 2) and a(0) = 1. - Armend Shabani, Mar 10 2017 a(n) = (Sum_{i=n-2..n+2} A000217(i))/5. a(n) = (Sum_{i=n-2..n+2} A002378(i))/10. a(n) = (Sum_{i=n..n+2} A002061(i)+1)/6. a(n) = (Sum_{i=n-1..n+2} A000290(i)+2)/8. (End)
Product_{n>=0} (1 + 1/a(n)) = cosh(sqrt(15)*Pi/2)*sech(sqrt(7)*Pi/2).
Product_{n>=1} (1 - 1/a(n)) = 2*Pi*sech(sqrt(7)*Pi/2). (End)
a((n^2-3n+6)/2) + a((n^2-n+4)/2) = a(n^2-2n+6)/2. - Charlie Marion, Feb 14 2023
EXAMPLE
a(3) = 7 because the 132- and 321-avoiding permutations of {1, 2, 3, 4} are 1234, 2134, 3124, 2314, 4123, 3412, 2341.
G.f. = 1 + 2*x + 4*x^2 + 7*x^3 + 11*x^4 + 16*x^5 + 22*x^6 + 29*x^7 + ...
MATHEMATICA
Table[PolygonalNumber[n] + 1, {n, 0, 52}] (* Michael De Vlieger, Jun 30 2016, Version 10.4 *)
PROG
(PARI) {a(n) = (n^2 + n) / 2 + 1}; /* Michael Somos, Sep 04 2006 */ (Haskell)
a000124 = (+ 1) . a000217
(Python)
def a(n): return n*(n+1)//2 + 1
CROSSREFS
Cf. A000096 (Maximal number of pieces that can be obtained by cutting an annulus with n cuts, for n >= 1). Cf. A002061, A002522, A016028, A055503, A072863, A144328, A177862, A263883, A000127, A005408, A006261, A016813, A058331, A080856, A086514, A161701, A161702, A161703, A161704, A161706, A161707, A161708, A161710, A161711, A161712, A161713, A161715, A051601, A228918. Cf. A055469 Quasi-triangular primes.
a(n) = 1 + (n-2)*(n-1)/2.
+10
29
1, 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, 172, 191, 211, 232, 254, 277, 301, 326, 352, 379, 407, 436, 466, 497, 529, 562, 596, 631, 667, 704, 742, 781, 821, 862, 904, 947, 991, 1036, 1082, 1129, 1177, 1226, 1276, 1327, 1379
COMMENTS
The sequence is the sum of upward sloping terms in an infinite lower triangle with 1's in the leftmost column and the odd integers in all other columns. - Gary W. Adamson, Jan 29 2014 For n > 1, if Kruskal's algorithm is run on a weighted connected graph of n nodes, then a(n) is the maximum number of iterations required to reach a spanning tree. - Eric M. Schmidt, Jun 04 2016 It can be observed that A152947/ A000079, whose reduced numerators are A213671, is identical to its inverse binomial transform (except for signs); this shows that it is an "autosequence" (more precisely, an autosequence of the second kind). - Jean-François Alcover (this remark is due to Paul Curtz), Jun 20 2016 Harnack's theorem (1876) states that the number of components of a plane algebraic curve of order n is at most a(n) and that this number can be achieved. For example, the zero set of a quadratic has at most 1 component (e.g. a circle); a cubic elliptic curve has at most 2 components. The number of topological arrangements (Hilbert's 16th problem) is only known for a few values of n. For quartics, a(4)=4 and there are 6 topological arrangements: 0 to 4 unnested ovals or 2 nested ovals. - Robert McLachlan, Aug 19 2024
FORMULA
E.g.f.: (4 - 2*x + x^2)*exp(x)/2 - 2.
Sum_{n>=1} 1/a(n) = 2*Pi*tanh(sqrt(7)*Pi/2)/sqrt(7) + 1 = A226985 + 1. (End)
PROG
(Sage) [1+binomial(n, 2) for n in range(0, 54)] # Zerinvary Lajos, Mar 12 2009
Decimal expansion of (Pi/2)*tanh(Pi/2).
+10
12
1, 4, 4, 0, 6, 5, 9, 5, 1, 9, 9, 7, 7, 5, 1, 4, 5, 9, 2, 6, 5, 8, 9, 3, 2, 5, 0, 2, 9, 1, 3, 9, 8, 1, 7, 1, 2, 5, 2, 5, 2, 9, 7, 0, 8, 4, 6, 7, 3, 6, 5, 8, 6, 9, 0, 8, 2, 1, 6, 1, 4, 0, 9, 2, 4, 6, 2, 0, 3, 1, 1, 5, 2, 2, 3, 3, 5, 6, 6, 0, 7, 7, 6, 4, 7, 9
COMMENTS
The old name was: Decimal expansion of sum of reciprocals, main diagonal of the natural number array, A185787. Let s(n) be the sum of reciprocals of the numbers in row n of the array T at A185787 given by T(n,k) = n + (n+k-2)(n+k-1)/2, and let r = (2*pi/sqrt(7))*tanh(pi*sqrt(7)/2), as at A226985. Then s(1) = r, and s(2) to s(5) are given by A228044 to A228047. Let c(n) be the sum of reciprocals of the numbers in column n of T. Then c(1) = 2; c(2) = 11/9, c(4) = 29/50, and c(3) is given by A228049. Let d(n) be the sum of reciprocals of the numbers in the main diagonal, (T(n,n)); then d(2) = (1/12)*(pi)^2; d(3) = 1/2, and d(1) is given by A228048. This is also the value of the series 1 + 2*Sum_{n>=1} 1/(4*n^4 + 1) = 1 + 2*(1/5 + 1/65 + 1/325 + ...). See the Koecher reference, p. 189. - Wolfdieter Lang, Oct 30 2017
REFERENCES
Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, p. 189.
FORMULA
Equals Integral_{x=0..oo} sin(x)/sinh(x) dx. - Amiram Eldar, Aug 10 2020 Equals Product_{k>=2} ((k^2 + 1)/(k^2 - 1))^((-1)^k). - Amiram Eldar, Apr 09 2022
EXAMPLE
1/1 + 1/5 + 1/13 + ... = (Pi/2)*tanh(Pi/2) = 1.4406595199775145926589...
MATHEMATICA
$MaxExtraPrecision = Infinity; t[n_, k_] := t[n, k] = n + (n + k - 2) (n + k - 1)/2; u = N[Sum[1/t[n, n], {n, 1, Infinity}], 130]; RealDigits[u, 10]
RealDigits[Pi*Tanh[Pi/2]/2, 10, 100][[1]] (* Amiram Eldar, Apr 09 2022 *)
Decimal expansion of sum of reciprocals, row 2 of the natural number array, A185787.
+10
6
1, 1, 2, 2, 2, 9, 4, 6, 0, 6, 6, 0, 3, 5, 0, 4, 3, 4, 3, 5, 4, 3, 4, 3, 2, 1, 8, 5, 9, 7, 9, 2, 5, 5, 9, 9, 2, 0, 2, 4, 3, 5, 0, 0, 8, 4, 2, 6, 9, 4, 6, 5, 5, 6, 7, 8, 8, 6, 4, 8, 1, 7, 3, 4, 3, 0, 8, 9, 9, 0, 3, 8, 1, 2, 1, 3, 5, 0, 3, 9, 6, 5, 8, 1, 0, 2
COMMENTS
Let s(n) be the sum of reciprocals of the numbers in row n of the array T at A185787 given by T(n,k) = n + (n+k-2)(n+k-1)/2, and let r = (2*pi/sqrt(7))*tanh(pi*sqrt(7)/2), as at A226985. Then s(1) = r, and s(2) to s(5) are given by A228044 to A228047. Let c(k) be the sum of reciprocals of the numbers in column k of T. Then c(1) = 2; c(2) = 11/9, c(4) = 29/50, and c(3) is given by A228049. Let d(n) be the sum of reciprocals of the numbers in the main diagonal, (T(n,n)); then d(2) = (1/12)*(pi)^2; d(3) = 1/2, and d(1) is given by A228048. It appears that Mathematica gives closed-form exact expressions for s(n), c(n) for 1<=n<=20 and further. The same holds for diagonal sums dr(n,n+k), k>=0; and for diagonal sums and dc(n+k,n), k>=0. In any case, general terms for all four sequences can be represented using the digamma function. The representations imply that c(n) is rational if and only if n is a term of A000124, and that dr(n) is rational if and only if n has the form k^2 + 2 for k >= 1.
EXAMPLE
1/3 + 1/5 + 1/8 + ... = (1/30)*(-15 + 8r*tanh(r/2), where r=(pi/2)sqrt(15).
1/3 + 1/5 + 1/8 + ... = 1.12229460660350434354343218597925...
MATHEMATICA
$MaxExtraPrecision = Infinity; t[n_, k_] := t[n, k] = n + (n + k - 2) (n + k - 1)/2;
u = N[Sum[1/t[2, k], {k, 1, Infinity}], 130]
RealDigits[u, 10]
Decimal expansion of sum of reciprocals, row 5 of the natural number array, A185787.
+10
6
4, 2, 3, 5, 3, 9, 0, 9, 9, 6, 0, 8, 7, 0, 0, 1, 9, 6, 8, 3, 7, 6, 0, 7, 6, 8, 9, 9, 7, 4, 4, 2, 8, 9, 3, 7, 5, 4, 4, 3, 2, 2, 8, 8, 1, 8, 9, 4, 1, 7, 1, 1, 1, 0, 2, 1, 7, 5, 6, 0, 8, 4, 2, 8, 1, 3, 0, 9, 3, 4, 7, 8, 2, 4, 5, 8, 2, 6, 7, 1, 1, 7, 8, 2, 5, 9
COMMENTS
Let s(n) be the sum of reciprocals of the numbers in row n of the array T at A185787 given by T(n,k) = n + (n+k-2)(n+k-1)/2, and let r = (2*pi/sqrt(7))*tanh(pi*sqrt(7)/2), as at A226985. Then s(1) = r, and s(2) to s(5) are given by A228044 to A228047. Let c(n) be the sum of reciprocals of the numbers in column n of T. Then c(1) = 2; c(2) = 11/9, c(4) = 29/50, and c(3) is given by A228049. Let d(n) be the sum of reciprocals of the numbers in the main diagonal, (T(n,n)); then d(2) = (1/12)*(pi)^2; d(3) = 1/2, and d(1) is given by A228048.
EXAMPLE
1/15 + 1/20 + 1/26 + ... = (1/17160)(-9997 + 1760r*tanh(r)), where r = (pi/2)*sqrt(39)
1/15 + 1/20 + 1/26 + ... = 0.42353909960870019683760768997442893...
MATHEMATICA
$MaxExtraPrecision = Infinity; t[n_, k_] := t[n, k] = n + (n + k - 2) (n + k - 1)/2; u = N[Sum[1/t[5, k], {k, 1, Infinity}], 130]; RealDigits[u, 10]
Decimal expansion of sum of reciprocals, column 3 of the natural number array, A185787.
+10
6
7, 9, 8, 4, 1, 0, 5, 5, 1, 0, 1, 6, 8, 7, 8, 0, 0, 3, 8, 6, 5, 2, 6, 6, 5, 1, 7, 5, 6, 1, 3, 2, 6, 5, 8, 1, 6, 6, 2, 7, 9, 3, 1, 6, 1, 9, 5, 4, 9, 8, 8, 5, 5, 7, 4, 1, 5, 2, 8, 6, 8, 7, 1, 8, 1, 1, 5, 7, 7, 8, 3, 0, 9, 5, 1, 4, 3, 1, 1, 1, 3, 3, 5, 4, 1, 9
COMMENTS
Let s(n) be the sum of reciprocals of the numbers in row n of the array T at A185787 given by T(n,k) = n + (n+k-2)(n+k-1)/2, and let r = (2*pi/sqrt(7))*tanh(pi*sqrt(7)/2), as at A226985. Then s(1) = r, and s(2) to s(5) are given by A228044 to A228047. Let c(n) be the sum of reciprocals of the numbers in column n of T. Then c(1) = 2; c(2) = 11/9, c(4) = 29/50, and c(3) is given by A228049. Let d(n) be the sum of reciprocals of the numbers in the main diagonal, (T(n,n)); then d(2) = (1/12)*(pi)^2; d(3) = 1/2, and d(1) is given by A228048.
EXAMPLE
1/4 + 1/8 + 1/13 + ... = (1/34)(17 + 8r*tan(r)), where r = (pi/2)sqrt(17)
1/4 + 1/8 + 1/13 + ... = 0.79841055101687800386526651756132658166...
MATHEMATICA
$MaxExtraPrecision = Infinity; t[n_, k_] := t[n, k] = n + (n + k - 2) (n + k - 1)/2; u = N[Sum[1/t[n, 3], {n, 1, Infinity}], 130]; RealDigits[u, 10]
Decimal expansion of 2*Pi*tanh(sqrt(5/3)*Pi/2)/sqrt(15).
+10
2
1, 5, 6, 7, 0, 6, 5, 1, 3, 1, 2, 6, 4, 0, 5, 4, 6, 7, 7, 5, 8, 8, 1, 1, 1, 5, 7, 7, 9, 5, 9, 9, 5, 4, 6, 4, 3, 9, 9, 5, 1, 6, 0, 0, 7, 3, 4, 7, 7, 6, 0, 2, 3, 0, 7, 4, 5, 4, 1, 2, 4, 3, 9, 8, 3, 1, 8, 4, 1, 0, 2, 0, 7, 0, 4, 1, 9, 8, 7, 6, 2, 5, 1, 5, 7, 4, 8, 4, 0, 6, 7, 0, 0, 3, 8, 0, 8, 3, 6, 1, 7, 7, 6, 9, 3, 0, 7, 6, 4, 0, 1, 3, 6, 2, 7, 6, 7, 9, 7, 9
COMMENTS
Decimal expansion of the sum of the reciprocals of the centered triangular numbers ( A005448).
FORMULA
Equals Sum_{k>=1} 1/(3*k*(k - 1)/2 + 1).
EXAMPLE
1.56706513126405467758811157795995464399516007...
MATHEMATICA
RealDigits[2 Pi Tanh[Sqrt[5/3] Pi/2]/Sqrt[15], 10, 120][[1]]
PROG
(PARI) 2*Pi*tanh(sqrt(5/3)*Pi/2)/sqrt(15) \\ Michel Marcus, Feb 08 2019
Decimal expansion of sum of reciprocals, row 3 of the natural number array, A185787.
+10
1
7, 2, 6, 8, 0, 0, 6, 1, 9, 4, 6, 4, 9, 3, 5, 7, 7, 8, 1, 7, 9, 1, 4, 3, 0, 0, 7, 1, 9, 4, 4, 3, 5, 3, 8, 3, 9, 0, 8, 7, 7, 4, 6, 2, 7, 6, 3, 6, 0, 7, 7, 7, 3, 2, 3, 8, 9, 9, 7, 2, 6, 1, 3, 4, 0, 1, 3, 4, 6, 7, 2, 7, 2, 0, 1, 4, 8, 5, 9, 5, 2, 6, 4, 2, 6, 4
COMMENTS
Let s(n) be the sum of reciprocals of the numbers in row n of the array T at A185787 given by T(n,k) = n + (n+k-2)(n+k-1)/2, and let r = (2*pi/sqrt(7))*tanh(pi*sqrt(7)/2), as at A226985. Then s(1) = r, and s(2) to s(5) are given by A228044 to A228047. Let c(n) be the sum of reciprocals of the numbers in column n of T. Then c(1) = 2; c(2) = 11/9, c(4) = 29/50, and c(3) is given by A228049. Let d(n) be the sum of reciprocals of the numbers in the main diagonal, (T(n,n)); then d(2) = (1/12)*(pi)^2; d(3) = 1/2, and d(1) is given by A228048.
EXAMPLE
1/6 + 1/9 + 1/13 + ... = (1/276)*(-161 + 48r*tanh(r/2), where r=(pi/2)sqrt(23).
1/6 + 1/9 + 1/13 + ... = 0.726800619464935778179143007194435...
MATHEMATICA
$MaxExtraPrecision = Infinity; t[n_, k_] := t[n, k] = n + (n + k - 2) (n + k - 1)/2; u = N[Sum[1/t[3, k], {k, 1, Infinity}], 130]; RealDigits[u, 10]
Decimal expansion of sum of reciprocals, row 4 of the natural number array, A185787.
+10
0
5, 3, 5, 6, 3, 6, 1, 9, 4, 7, 8, 0, 7, 8, 7, 2, 8, 4, 5, 5, 7, 8, 5, 0, 7, 4, 8, 6, 6, 4, 7, 1, 8, 6, 0, 7, 0, 0, 4, 3, 5, 4, 9, 4, 9, 6, 9, 1, 0, 7, 9, 6, 2, 2, 7, 7, 5, 5, 2, 0, 9, 8, 4, 9, 1, 8, 8, 7, 9, 3, 3, 8, 3, 3, 6, 0, 7, 1, 3, 2, 4, 9, 7, 9, 7, 8
COMMENTS
Let s(n) be the sum of reciprocals of the numbers in row n of the array T at A185787 given by T(n,k) = n + (n+k-2)(n+k-1)/2, and let r = (2*pi/sqrt(7))*tanh(pi*sqrt(7)/2), as at A226985. Then s(1) = r, and s(2) to s(5) are given by A228044 to A228047. Let c(n) be the sum of reciprocals of the numbers in column n of T. Then c(1) = 2; c(2) = 11/9, c(4) = 29/50, and c(3) is given by A228049. Let d(n) be the sum of reciprocals of the numbers in the main diagonal, (T(n,n)); then d(2) = (1/12)*(pi)^2; d(3) = 1/2, and d(1) is given by A228048.
EXAMPLE
1/10 + 1/14 + 1/19 + ... = (1/4340)*(-2573 + 560r*tanh(r/2), where r=(pi/2)sqrt(31)
1/10 + 1/14 + 1/19 + ... = 0.5356361947807872845578507486647...
MATHEMATICA
$MaxExtraPrecision = Infinity; t[n_, k_] := t[n, k] = n + (n + k - 2) (n + k - 1)/2; u = N[Sum[1/t[4, k], {k, 1, Infinity}], 130]; RealDigits[u, 10]
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